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Relevance heuristics allow us to tailor a program analysis to a particular property to be verified. This in turn makes it possible to improve the precision of the analysis where needed, while maintaining scalability. In this talk I will discuss the principles by which SAT solvers and other decision procedures decide what information is relevant to a given proof. Then we will see how these ideas can be exploited in program verification using the method of Craig interpolation. The result is an analysis that is finely tuned to prove a given property of a program. At the end of the talk, I will cover some recent research in this area, including the use of interpolants for verifying heap-manipulating programs.